It is a classical result of Raikov and Gelfand that if $G$ is a locally compact abelian group and $\phi:G\rightarrow\mathbb{C}$ is a continuous positive definite function, then there exists a representation $U$ of $G$ on Hilbert space $\mathcal{H}$ and a cyclic vector $f \in \mathcal{H}$ for which $\phi(g) = \langle U_gf,f\rangle$. Through the use of the Gaussian Measure Space Construction (GMSC), the previous result can be refined by taking $\mathcal{H} = L^2(X,\mu)$ and letting $U_g$ be the Koopman representation of a measure preserving action of $G$. Our first main result is to further refine the latter result by showing that the measure preserving action of $G$ can be assumed to be ergodic. Our second main result is when $G$ is abelian, in which case we refine the result of the GMSC in a direction by showing that $f \in L^2(X,\mu)$ can be taken to satisfy $|f| = 1$ a.e. We will also review a classical result of Foias and Stratila that shows that we cannot always take the system in the previous result to be ergodic. If time permits, we will discuss connections to the study of van der Corput sets.